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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Centroid of connected components</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">This algorithm is based on segmentation of molecules which is performed
by applying a user-defined <a href="Threshold.html" title="" class="ltx_ref">threshold</a> on the
filtered image <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi2.png" width="18" height="16" alt="F">. Users also have the option to combine the thresholded
image with the result of a watershed algorithm <cite class="ltx_cite">[<a href="#bib.bib1" title="Image Processing, Analysis, and Machine Vision" class="ltx_ref">3</a>]</cite> applied
to the image <img id="Sx1.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi2.png" width="18" height="16" alt="F">. This procedure was proposed in <cite class="ltx_cite">[<a href="#bib.bib16" title="Wavelet analysis for single molecule localization microscopy" class="ltx_ref">1</a>]</cite>.
The position of each molecule is determined by computing the centroid
of the relevant segmented object using</p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-25px" src="mi/mi1.png" width="278" height="57" alt="\tilde{x}_{p}=\frac{1}{c_{p}}\sum_{i=1}^{c_{p}}{x_{i,p}}\,,\quad\tilde{y}_{p}=%
\frac{1}{c_{p}}\sum_{i=1}^{c_{p}}{y_{i,p}}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
<p class="ltx_p">Here <img id="Sx1.p1.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi4.png" width="13" height="15" alt="p"> indexes the objects, <img id="Sx1.p1.m4" class="ltx_Math" style="vertical-align:-7px" src="mi/mi3.png" width="20" height="17" alt="c_{p}"> is the number of elements
in each object, and <img id="Sx1.p1.m5" class="ltx_Math" style="vertical-align:-7px" src="mi/mi5.png" width="32" height="17" alt="x_{i,p}"> and <img id="Sx1.p1.m6" class="ltx_Math" style="vertical-align:-7px" src="mi/mi6.png" width="30" height="17" alt="y_{i,p}"> are the integer pixel
coordinates within each object. Segmented objects are determined as
connected components by an algorithm based on a breadth-first search
<cite class="ltx_cite">[<a href="#bib.bib19" title="The Art Of Computer Programming" class="ltx_ref">2</a>]</cite>. The watershed algorithm is based on the ImageJ
‘‘Find Maxima’’ function.</p>
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<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Detectors.html" title="" class="ltx_ref">Finding approximate positions of molecules</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="Threshold.html" title="" class="ltx_ref">Threshold selection</a></p>
</div>
</li>
</ul>
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<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib16" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">I. Izeddin, J. Boulanger, V. Racine, C. G. Specht, A. Kechkar, D. Nair, A. Triller, D. Choquet, M. Dahan and J. B. Sibarita</span><span class="ltx_text ltx_bib_year">(2012)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Wavelet analysis for single molecule localization microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Optics Express</span> <span class="ltx_text ltx_bib_volume">20</span> (<span class="ltx_text ltx_bib_number">3</span>), <span class="ltx_text ltx_bib_pages"> pp. 2081–95</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/OE.20.002081" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Centroid of connected components" class="ltx_ref"><span class="ltx_text ltx_ref_title">Centroid of connected components</span></a>.
</span>
</li>
<li id="bib.bib19" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">D. E. Knuth</span><span class="ltx_text ltx_bib_year">(1997)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">The Art Of Computer Programming</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_edition">3rd edition</span>, Vol. <span class="ltx_text ltx_bib_volume">1</span>,  <span class="ltx_text ltx_bib_publisher">Addison-Wesley</span>, <span class="ltx_text ltx_bib_place">Boston</span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Centroid of connected components" class="ltx_ref"><span class="ltx_text ltx_ref_title">Centroid of connected components</span></a>.
</span>
</li>
<li id="bib.bib1" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[3]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">M. Šonka, V. Hlaváč and R. Boyle</span><span class="ltx_text ltx_bib_year">(2007)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Image Processing, Analysis, and Machine Vision</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_edition">3rd edition edition</span>,  <span class="ltx_text ltx_bib_publisher">Cengage Learning</span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Centroid of connected components" class="ltx_ref"><span class="ltx_text ltx_ref_title">Centroid of connected components</span></a>.
</span>
</li>
</ul>
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